Babies and toddlers (and older children too) need shoes that protect their feet from sharp objects and offer a bit of warmth when it's cold. The most important things are that the shoes offer a fully flexible sole that bends with the foot, and a wide enough toe box for the toes to spread out when walking.
They. Do. Not. Need. Support.
(Unless there are specific orthopaedic problems).
Stiff-soled shoes that make the wearer clomp around rather than walk by flexing the foot are bad for foot development and posture. Built-up boots that prevent the ankle from flexing are the same. Shoes shouldn't interfere with walking: a child should have the same gait barefoot and shod.
Clarks sell some good, flexible shoes and some awful, stiff shoes. Strict width fittings are sold as being important in over-structured, stiff shoes because they aim to encase the foot and leave it little ability to move. Width fittings aren't so important in minimal, flexible shoes because they aim to allow the foot to spread out and move naturally.
Lots of companies sell excellent flexible shoes, sometimes called barefoot shoes as they aim to encourage the same style of walking that children naturally adopt when barefoot - Bobux, Vivobarefoot, Livie and Luca, some Pediped, etc. All of these shoes can be fitted by measuring at home. Only unusually narrow or wide feet might need some advice about which brand might suit best - as a pp linked earlier, happylittlesoles.com are great for this.